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The 7th Hermite Polynomial evaluated at n: H_7(n) = 128*n^7 -1344*n^5 + 3360*n^3 - 1680*n.
2

%I #23 Feb 16 2025 08:33:23

%S 0,464,-3104,39024,929216,6211600,26096544,83965616,226102144,

%T 535292496,1148943200,2282359024,4257827136,7540152464,12779289376,

%U 20860714800,32964187904,50631541456,75844149984,111110719856,159566046400,225081383184,312387068576

%N The 7th Hermite Polynomial evaluated at n: H_7(n) = 128*n^7 -1344*n^5 + 3360*n^3 - 1680*n.

%H Vincenzo Librandi, <a href="/A247852/b247852.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HermitePolynomial.html">Hermite Polynomial</a>.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F G.f.: x*(464-6816*x+76848*x^2+504128*x^3+76848*x^4-6816*x^5 +464*x^6)/(1-x)^8.

%F a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8).

%t Table[128 n^7 - 1344 n^5 + 3360 n^3 - 1680 n, {n, 0, 30}] (* or *) CoefficientList[Series[x (464 - 6816 x + 76848 x^2 + 504128 x^3 + 76848 x^4 - 6816 x^5 + 464 x^6)/(1-x)^8, {x, 0, 30}], x]

%o (Magma) [128*n^7-1344*n^5+3360*n^3-1680*n: n in [0..30]];

%o (Magma) I:=[0,464,-3104,39024,929216,6211600,26096544,83965616]; [n le 8 select I[n] else 8*Self(n-1)-28*Self(n-2)+56*Self(n-3)-70*Self(n-4)+56*Self(n-5)-28*Self(n-6)+8*Self(n-7)-Self(n-8): n in [1..30]];

%o (PARI) a(n)=polhermite(7,n) \\ _Charles R Greathouse IV_, Jan 29 2016

%o (Python)

%o from sympy import hermite

%o def A247852(n): return hermite(7, n) # _Chai Wah Wu_, Jan 06 2022

%Y Cf. similar sequences listed in A247850.

%K sign,easy,changed

%O 0,2

%A _Vincenzo Librandi_, Sep 25 2014